Mathematics, 01.08.2021 01:00, jakeyywashere

A) Show that the equation 2 x + 3 cos x + e^x = 0 has a root on the interval (-1,0)
b) Use the Bisection method to find the solution of 2 x + 3 cos x +
e^x = 0. accurate with in 10-3. On (-1,0). (Use four digits-Rounding)

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answered: angelb2472

a) The equation has a root in the interval (-1,0)

b) The solution of $2x+3cos(x)+e^{x}=0$ by using the Bisection Method is x=0.9977 accurate within $10^{-3}$

a) The intermediate zero theorem (Bolzano's Theorem) tells us that whenever you have a continuous function in a given interval and the extremes of the functions on this interval have oposite signs, then there must be a zero in between those extreme values.

A formal definition of this theorem is written like this:

"If a function f on the closed interval [a,b] is a continuous function and it holds that f(a)>0 and f(b)<0 or f(a)<0 and f(b)>0, then there is at least one x-value such that f(x)=0"

So basically we need to evaluate the given equation for both extremes of the interval x=-1 and x=0, if they return results opposite in sign, then there must be a zero in that interval, so let's evaluate the function for x=-1:

$2x+3cos(x)+e^{x}$

$2(-1)+3cos(-1)+e^{-1}=-0.0112$

Let's now test for x=0

$2x+3cos(x)+e^{x}$

$2(0)+3cos(0)+e^{0}=4$

So notice we ended up with two values -0.0112 and 4. One is positive and the other is negative, therefore there must be a zero in that interval.

b) The zero is located at x=-0.9977

The idea of the bisection method is to find values for x in the middle of two x-values that return opposite sign answers when evaluated on the given function. So we can start with the extremes of the given interval:

x=-1 and x=0

so we find the value in the middle by using the following formula:

$mid-value=\frac{x_{1}+x_{2}}{2}$

so we get:

$mid-value=\frac{-1+0}{2}$

mid-value=-0.5

Next, we evaluate the given function for that value:

$2(-0.5)+3cos(-0.5)+e^{-0.5}=2.2393$

Since we got a positive answer, we now find the midpoint between -0.5 and -1 (which was the last x-value that returned a negative answer) so we get:

$mid-value=\frac{-1-0.5}{2}$

mid-value=-0.75

Next, we evaluate the given function for that value:

$2(-0.75)+3cos(-0.75)+e^{-0.75}=1.1674$

and we repeat the process until que get to the desired accuracy. I uploaded a table that has the corresponding iterations and its answers. There were 14 iterations done until we got to the final answer x=-0.9977.

Learn more about the intermediate zero theorem here: